What does integrability of finite-gap or soliton potentials mean?

@article{Brezhnev2008WhatDI,
title={What does integrability of finite-gap or soliton potentials mean?},
author={Yurii V. Brezhnev},
journal={Philosophical transactions. Series A, Mathematical, physical, and engineering sciences},
year={2008},
volume={366 1867},
pages={923-45}
}

In the example of the Schrödinger/KdV equation, we treat the theory as equivalence of two concepts of Liouvillian integrability: quadrature integrability of linear differential equations with a parameter (spectral problem) and Liouville's integrability of finite-dimensional Hamiltonian systems (stationary KdV equations). Three key objects in this field-new explicit Psi-function, trace formula and the Jacobi problem-provide a complete solution. The Theta-function language is derivable from these… CONTINUE READING